Size Scaling Law for Radiation Losses of Modes in Photonic Crystal Surface Emitting Devices

Abstract

Photonic-crystal surface-emitting lasers (PCSELs) have garnered significant attention due to their ability to generate laser beams with ultra high power and low divergence. This is because they support high power single mode lasing with volumes orders of magnitude larger than those of conventional semiconductor lasers. The finite lateral size in a PCSEL is a primary factor limiting its lasing mode volume and consequently, its output power. We demonstrate that the scaling relation between the total cavity loss $α=α_\perp + α_\parallel$ and the device size $L$ is such that the surface radiation loss scales as $α_{\perp} \sim O(L^{-2})$, while the edge radiation loss $α_{\parallel} \sim O(L^{-3})$. Both scaling relations can be explained by the second order expansions of the complex frequency $ω(k)$ of the band diagram. Our results constitute an explicit guideline for PCSEL designs to optimize various optical properties.

Figures (8)

• Figure 1: The schematic of the structure of a PCSEL. The red and the orange arrows denote the surface $\alpha_\perp$ and the edge radiation loss $\alpha_\parallel$, respectively. The gray area represents the non-PhC layers in the device, that may include cladding layers, gain media layers, etc. The three insets show the simulated unit cell structure with circular, isosceles right-triangle, and double-lattice holes. The double-lattice unit cell consists of both elliptical and circular holes.
• Figure 2: Comparison of the band structure calculated using 3D-CWT (solid lines) and Finite Element Modeling (FEM, dotted lines) for modes $\mathrm{A}$ and $\mathrm{B}$. The insets show the magnitude of the magnetic field (in color) and the direction of the electric field (indicated by black arrows) within a unit cell for modes $\mathrm{A}$ and $\mathrm{B}$, respectively. The plot is calculated for circular unit cell with filling factor $f=0.12$.
• Figure 3: Mode spectrum. (a) The mode spectrum of a finite-size PCSEL with $L=400a$. Resonance modes $\mathrm{A}$ and $\mathrm{B}$ are marked by red and blue dashed circles, respectively. The basic modes $\mathrm{A_0}$ and $\mathrm{B_0}$, as well as the first higher-order modes $\mathrm{A_1}$ and $\mathrm{B_1}$, are indicated with arrows. Data points highlighted in yellow are nonphysical artifacts arising from discretizationsRN148. The solid red and blue lines depict the parametric plot of $\delta(k)-\alpha(k)$ for modes $\mathrm{A}$ and $\mathrm{B}$, obtained via Eq. (\ref{['eq:finite_equation_with_k']}). (b) The mode spectrum after eliminating edge radiation loss for each mode in (a). (c), (d) The magnitude of the magnetic field (in color) and the direction of the electric field (indicated by black arrows) within a unit cell for modes $\mathrm{A}$ and $\mathrm{B}$, respectively. The field intensity envelopes for modes labeled $\mathrm{A_0}, \mathrm{A_1}, \mathrm{B_0}$ and $\mathrm{B_1}$.
• Figure 4: The surface radiation loss ($\alpha_{\perp}$) and the edge radiation loss ($\alpha_{\parallel}$) for PCSEL devices with circular unit cells (the inset) in square lattices of varying device size $L$. (a)$\mathrm{A_0}$ mode. (b)$\mathrm{B_0}$ mode. The dashed lines are the linear fits for data points with $L>1000a$.
• Figure 5: The radiation loss of the $\mathrm{A_0}$ and $\mathrm{B_0}$ modes for PCSELs with different unit cell structure. (a), (b) isosceles right-triangular and (c), (d) double-lattice, see the inset. $\alpha_\infty$ is the surface radiation for $L\to\infty$ of the same structure.